HEAVISIDE FUNCTION IN PRECALCULUS (WITH SOLVED PROBLEMS)

Introduction to the Heaviside Function

  • Invented by Oliver Heaviside, a self-taught mathematician, physicist, and electrical engineer.

  • Denoted typically by a capital letter H.

  • Small letter h refers to a general function, not the Heaviside function.

Characteristics of the Heaviside Function

  • Returns only two values: 0 or 1.

    • If the input value (x) is less than zero: returns 0.

    • If the input value (x) is equal to zero: returns 0.

    • If the input value (x) is greater than zero: returns 1.

  • Can be applied in various fields, including:

    • Laplace transformation.

    • Electrical devices (used to define on/off states).

    • Medical devices for monitoring body functions.

Examples of the Heaviside Function

  • H(23) returns 1 (23 > 0).

  • H(0) returns 0.

  • H(-15.2) returns 0.

  • H(7.38) returns 1.

  • H(1.35) returns 1.

  • H(1) returns 1.

  • H(-1) returns 0.

  • H(-5) returns 0.

Important Notes

  • Remember to denote the Heaviside function properly; using a small h is incorrect.

  • In exam scenarios, problems may involve substituting x for specific values to find answers based on the Heaviside function.

Example Problems

  • For an exam question:

    • Replace x with 7: H(7) returns 1, with an additional answer to retrieve, such as 5/2.

  • Combine Heaviside function with other functions:

    • Signum() function and Identity function may be presented in problems.

    • Example: Signum(-25) = -1, H(3) = 1, Identity(-7) = -7. In the calculation: 9 - 1 - 1 = -2, and -7 + 9 = 2; thus the answer is -1.

Conclusion

  • The identity function is now introduced.

  • Further learning can be pursued through additional video resources.

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